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Ajman

Vectors And Transformations

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Published in: Mathematics
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Representing translation by a vector Add and subtract vectors Multiply a vector by a scalar

Divya J / Dubai

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Qualification: CAIE and Pearson Examiner for IGCSE an AS Levels, Masters in Mathematics

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  1. Subject - Mathematics 01 Lecture No I-I Chapter • Vectors and Transformations Today's Targets Representing translation by a vector Add and subtract vectors Multiply a vector by a scalar
  2. Vector notation Vectors can be represented by a directed line segment as shown in the diagrams on the right. a B
  3. Using vectors to describe a translation A translation (a sliding movement) can be described using column Vectors. A column vector describes the movement of the object in both the x direction and they direction.
  4. Using vectors to describe a translation The diagrammatic representation of -a and —b is shown below
  5. Using vectors to describe a translation Addition and subtraction of vectors: Vectors can be added together and represented diagrammatically as shown.
  6. Multiplying a vector by a scalar (0 by a scalar k, gives k Multiplying any vector Vectors cannot be multiplied by each other, but they can be multiplied by a constant factor or scalar. Vector a multiplied by 2 is the vector 2a. Vector 2a is twice as long as vector a, but they have the same direction. In other words, they are either parallel or in a straight line.
  7. Question Describe the translation from A to B in the diagram in terms of a column vector.
  8. Question Describe each of the translations shown using column vectors.
  9. Question (a) Draw a diagram to represent a - b, where a - b = (a) + (—b).
  10. Question (b) Calculate the vector represented by a - b.
  11. Question If a = , express the vectors b, c, d and e in terms of a.
  12. Question In the following, Calculate: 2a + b
  13. [M20, 22. Q3] Point A has coordinates (6, 4) and point B has coordinates (2, 7). Write AB as a column vector. AB = ( ) [11
  14. [S20, 21, Q17a(?)] Find 3m.
  15. [S20, 42, Q2a] Find 2P + q.
  16. AB = Find [W20, 43. Q8a(i), a(ii)] (-23) AC = BD = AC = ( ) 12] BD = ( ) [21
  17. Revision • A vector is a type of number that has both a size and a direction. In transformation geometry, translations are indicated in the form of a column vector: Movement in x — Direction (+ To right, - To left) x Movement in y-Direction (+ UP, - Down)
  18. Revision Note in particular that vector -a is the same size as vector a, but points in the opposite direction! When vectors are represented as column vectors, adding or subtracting is simply a matter of adding or subtracting the vectors' x and y coordinates Note that multiplying by a negative scalar also changes the direction of the vector
  19. Revision Think of a Vector as a Journey AB = a 1. 2. Although a VECTOR may be defined as going from one point to another it can be used anywhere - (32) = To add/subtract multiples of COLUMN VECTORS treat x and y components separately.

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